Another interesting approach to circumventing the computational barriers to computing pure equilibrium in congestion games considers beyond worst case analysis. Specifically a recent line of works applies smoothed analysis to the special case of polymatrix identical interest games [1]}, [2]}, [3]}, [4]}, [5]}. It is an interesting open question whether considering mixed Nash equilibrium can improve those results. Another interesting question is whether this analysis can extend to more general congestion games (for an appropriate definition of smoothing).
